Random Matrices: Beyond Wigner and Marchenko-Pastur Laws

Transcription

1 Random Matrices: Beyond Wigner and Marchenko-Pastur Laws Nathan Noiry Modal X, Université Paris Nanterre May 3, 2018

2 Wigner s Matrices Wishart s Matrices Generalizations

3 Wigner s Matrices ij, (n, i, j 1): collection of i.i.d. random variables such that E[X ] = 0 and E[X 2 ] = 1. For all n 1: W n = 1 n n n. 1n n X nn (n) Symmetry: n real eigenvalues λ (n) 1 λ (n) 2 λ (n) n.

4 Simulations Gaussian case: X N (0, 1), 100 matrices of size

5 Simulations Gaussian case: X N (0, 1), 100 matrices of size σ(dx) = 4 x 2 1 2π x 2 dx

6 Convergence to the semi-circle law Empirical spectral measure: µ Wn = 1 n n i=1 δ (n) λ. i Theorem (Wigner, 1958) Almost surely, µ Wn weakly converges to σ: for all bounded measurable function f : R R +, R f (x)dµ Wn (x) = 1 n n ( f i=1 λ (n) i ) a.s. n x 2 f (x) dx. 2π

7 Wishart s Matrices m = αn, α > 0; for all n 1, rectangular n m matrix: R n = m m. n n2 X nm (n) Wishart s matrix of size n: covariance matrix C n = 1 n R nr T n. Positive matrix: n positive eigenvalues λ (n) 1 λ (n) n.

8 Simulations Gaussian case: X N (0, 1), 100 matrices of size (α = 2)

9 Convergence to the Marchenko-Pastur law MP α (dx) = (b x)(x a) 1 2πx (a,b) (x)dx + (1 α)1 α<1 δ 0 (dx). a = (1 α) 2, b = (1 + α) 2. α = 2: α = 4:

10 Convergence to the Marchenko-Pastur law Empirical spectral measure: µ Cn = 1 n n i=1 δ (n) λ. i Theorem (Marchenko-Pastur, 1967) Almost surely, µ Wn weakly converges to MP α : for all bounded measurable function f : R R +, R f (x)dµ Wn (x) = 1 n n ( f i=1 λ (n) i ) a.s. n + R f (x)dmp α (x).

11 Wigner case: Generalization {P n }: sequence of probability laws, M 2 (P n ) = 1. Symmetric matrix: ij s i.d.d. with law P n. W n = 1 n ( ij ) 1 i,j n, Convergence of spectral empirical measures µ Wn : Ryan (1998); Zakharevich (2006).

12 Wishart case: Generalization {P n }: sequence of probability laws, M 2 (P n ) = 1. ( ) R n = M n m, ij m = αn; ij i.d.d. with law P n. C n = 1 n R nr T n. Convergence of spectral empirical measures µ Cn : Benaych-George and Cabanal-Duvillard (2012); N. (2017).

13 Theorem (N.,2017) If M k (P n ) n k/2 1 A k <, n + the weak limit of µ Cn only depends on (A i ) i and is characterized by its moments: m k = k a a=1 l=1 α l b B a,k W k (a, l, b) a A bi. i=1 W k (a, l, b): closed walks on planar rooted trees having a edges and l vertices in odd generations, with repetitions given by b.

14 Combinatorial analysis of the formula m k = k l=1 α l k 1 W k (k, l, (2,..., 2)) + A 4 α l W k (k 1, l, (4, 2,..., 2)) + l=1

15 Combinatorial analysis of the formula m k = k α l k 1 W k (k, l, (2,..., 2)) + A 4 l=1 l=1 ( m k = m k (MP α) + A 4m k MP {1} α α l W k (k 1, l, (4, 2,..., 2)) + ) +

16 Combinatorial analysis of the formula m k = k α l k 1 W k (k, l, (2,..., 2)) + A 4 l=1 l=1 ( m k = m k (MP α) + A 4m k MP {1} α α l W k (k 1, l, (4, 2,..., 2)) + ) + MP {1} α : explicit signed measure with total mass zero. α (dx) = x 2 2x(α + 1) + (α 2 + 1) 2πα 1 (a,b) (x)dx. (b x)(x a) MP {1}

17 Combinatorial analysis of the formula m k = k α l k 1 W k (k, l, (2,..., 2)) + A 4 l=1 l=1 ( m k = m k (MP α) + A 4m k MP {1} α α l W k (k 1, l, (4, 2,..., 2)) + ) + MP {1} α : explicit signed measure with total mass zero. α (dx) = x 2 2x(α + 1) + (α 2 + 1) 2πα 1 (a,b) (x)dx. (b x)(x a) MP {1} One example of application: P n = Bernoulli ( ) c n where Ai = c 1 i/2. Asymptotic development of the limiting law µ α,c when c + : m k (µ α,c) = m k (MP α) + 1 c m k ( MP {1} α ) + o ( 1 c ).

18 Wigner s Matrices Wishart s Matrices 2.5 Generalizations THANK 0.5 YOU! Reference: Spectra of Wishart s Matrices with Size-Dependent Entries, N., 2017 (preprint).